COST 273, Bologna meeting
KEYHOLES AND MIMO CHANNEL
MODELLING
Alain SIBILLE
[email protected]
ENSTA
32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE
Outline
 Keyholes in MIMO channels viewed as the result of diffraction
Outline
 Keyholes in MIMO channels viewed as the result of diffraction
 Channel modelling with multipath junctions in a small
Size, uncoupled sensors antenna approximation
Outline
 Keyholes in MIMO channels viewed as the result of diffraction
 Channel modelling with multipath junctions in a small
Size, uncoupled sensors antenna approximation
 How to include inter-sensors coupling
Outline
 Keyholes in MIMO channels viewed as the result of diffraction
 Channel modelling with multipath junctions in a small
Size, uncoupled sensors antenna approximation
 How to include inter-sensors coupling
 towards a stochastic MIMO channel model
Outline
 Keyholes in MIMO channels viewed as the result of diffraction
 Channel modelling with multipath junctions in a small
Size, uncoupled sensors antenna approximation
 How to include inter-sensors coupling
 towards a stochastic MIMO channel model
 Conclusion
Keyholes :
The concept of « Keyholes » has been suggested by Chizhik in order to hightlight the
imperfect correspondence between rank and correlation. In a keyhole, the channel
matrix has uncorrelated entries, but its rank is one. Such keyholes have therefore
intrinsically a small capacity, even in a rich scattering environment.
Slit transmittance
 B1 
 
H   B2  K  A1
B 
 3
A2
 A1 B1

A3   K  A1 B2
AB
 1 3
1D channel
A2 B1
A2 B2
A1 B3
A3 B1 

A3 B2 
A3 B3 
B2
A1
B1
B3
K
A2
A3
Rank(H)=1 (two null coefficients of characteristic polynomial)
E ( A1B1 A3 B2 )  E ( A1B1 ) E ( A3 B2 )  0
: uncorrelated (complex) entries
Keyholes in MIMO channels
A simple numerical example of keyhole using Kirchhoff diffraction:
Rx
Tx
Large slit: no diffraction
Keyholes in MIMO channels
A simple numerical example of keyhole using Kirchhoff diffraction:
junction
Rx
Large slit: no diffraction
 
H ( , rr , rt ) 
Tx
Narrow slit: diffraction and
multipath junction 1  3
 
 
 Ri .T j .K ij exp(  jkli ) exp(  jkl j ) exp(  jki  rr ) exp(  jk j  rt )
3
i 1, j 1
Kij computed by Kirchhoff diffraction
Keyholes in MIMO channels
H: (normalized) channel transmission
matrix
cumulated probability
nt=3: number of Tx, Rx radiators
SNR = 3 dB
0.8
Space-variant stochastic ensemble
0.6
 

SNR
C  log 2 det I nr 
H .H  
nt

 
0.4
case A
wide slit
little correlation : 3 DF
SV: -7, +4.8, +7.6 dB
0.2
0
2.5
3
capacity (b/s/Hz)
3.5
4
Keyholes in MIMO channels
B: narrow slit, little
correlation : 1 DF
SV: -47, -28, +9.5 dB
cumulated probability
case B
0.8
0.6
0.4
case A
wide slit
little correlation : 3 DF
SV: -7, +4.8, +7.6 dB
0.2
0
2.5
3
capacity (b/s/Hz)
3.5
4
Keyholes in MIMO channels
C: narrow slit, strong
correlation : 1 DF
SV: -111, -41, +9.5 dB
cumulated probability
1
B: narrow slit, little
correlation : 1 DF
SV: -47, -28, +9.5 dB
case B
case C
0.8
0.6
0.4
case A
wide slit
little correlation : 3 DF
SV: -7, +4.8, +7.6 dB
0.2
0
2.5
3
capacity (b/s/Hz)
3.5
4
Keyholes in MIMO channels: capacity vs. Slit width
cumulated probability
1
2
0.8
5
Slit width in units of l
0.6
0.4
0.2
0
2
2.5
3
capacity (b/s/Hz)
3.5
4
Keyholes in MIMO channels: capacity vs. Slit width
cumulated probability
1
0.8
0.25
0.5
2
5
Slit width in units of l
0.6
0.4
0.2
0
2
2.5
3
capacity (b/s/Hz)
3.5
4
Keyholes in MIMO channels: capacity vs. Slit width
cumulated probability
1
0.8
0.25
1
0.5
2
5
Slit width in units of l
0.6
0.4
0.2
0
2
2.5
3
capacity (b/s/Hz)
3.5
4
Keyholes in MIMO channels: capacity vs. Slit width
cumulated probability
1
0.8
0.25
1
0.5
2
5
Slit width in units of l
0.6
0.4
0.2
0
2
2.5
3
3.5
4
capacity (b/s/Hz)
When d< ~ l/2 all incoming waves are diffracted into all exiting waves through a
1-dimensional channel
Keyholes in MIMO channels: capacity vs. Slit width
cumulated probability
1
0.8
0.25
1
0.5
2
5
Slit width in units of l
0.6
0.4
0.2
0
2
2.5
3
3.5
4
capacity (b/s/Hz)
When d< ~ l/2 all incoming waves are diffracted into all exiting waves through a
1-dimensional channel
When d>~2l transmission through the slit occurs through multiple modes and
evanescent states and results in greater 3 dimensional effective channel
Keyholes : correlations or no correlations ?
 B1 
 
H   B2  K  A1
B 
 3
A2
 A1 B1

A3   K  A1 B2
AB
 1 3
A2 B1
A2 B2
A1 B3
A3 B1 

A3 B2 
A3 B3 
B2
Rank(H)=1 (two null coefficients of characteristic polynomial)
E ( A1B1 A3 B2 )  E ( A1B1 ) E ( A3 B2 )  0
A1
B1
B3
K
A2
A3
junction
: uncorrelated (complex) entries
Keyholes : correlations or no correlations ?
 B1 
 
H   B2  K  A1
B 
 3
 A1 B1

A3   K  A1 B2
AB
 1 3
A2
A2 B1
A2 B2
A1 B3
A3 B1 

A3 B2 
A3 B3 
B2
B3
Rank(H)=1 (two null coefficients of characteristic polynomial)
E ( A1B1 A3 B2 )  E ( A1B1 ) E ( A3 B2 )  0
   E  A    0
2
2
1
K
A2
A3
junction
: uncorrelated (complex) entries
E  A1 B1 . A1 B2   E  A1 B1 E  A1 B2 
 E  B1 E  B2  E A1
A1
B1
: correlated amplitudes
Keyholes in MIMO channels
B: narrow slit, little
correlation : 1 DF
SV: -47, -28, +9.5 dB
C: narrow slit, strong
correlation : 1 DF
SV: -111, -41, +9.5 dB
1
case B
case C
cumulated probability
0.8
case D
0.6
narrow slit, strong
correlation, one random
phase: 2 DF
SV: -40, -3.9, +9.4 dB
0.4
case A
wide slit
little correlation : 3 DF
SV: -7, +4.8, +7.6 dB
0.2
fading
0
2.5
3
capacity (b/s/Hz)
3.5
4
MIMO channel modelling
Small antenna, uncoupled sensors approximation:
{
H( )  a r ( ) W( )a Tt ( )
H()ar()W()aTt()
ar , at : steering matrices for N DOAs and M DODs (nrXN , MXnt )
W : wave connecting matrix (NXM) : complex attenuations from all DODs to
all DOAs
W is in general rectangular in the presence of path junctions (diffraction,
refraction …)
DOD
Rank(H)  Min(nr , nt , N , M)

DOA
All MIMO properties determined
by the geometry of sensors and
by W (DOD, DOA, complex amplitudes)
X

0
0

0
0

0
0
0
0
X
0
0
0
0
X
X
0
0
0
0
X
0
0
0
X
0

X
0

0
0 
MIMO channel modelling
Rx
m
Example: channel correlation matrix
(US approximation, spatial averaging)

R mnmn   Wi ,i
i ,i 

n’
m’

R  E a r ( ) W( )a Tt ( )  a*r ( ) W * ( )a tH ( )
2
Tx
n

 

2



 exp  j  k ri  rrm  rrm   j  k ti  rtn  rtn  /  Wi ,i
DOA
Receiver sensors positions
i ,i 
DOD
Transmitter radiators positions
MIMO channel modelling : case of coupled sensors
Uncoupled sensors

ki
Steering matrix ar (nrXN)
 
a r,i, j  exp(  jk j  rri )
H( )  a r ( ) W( )a Tt ( )
MIMO channel modelling : case of coupled sensors
Uncoupled sensors
Coupled sensors

ki
Steering matrix ar (nrXN)
Complex gain matrix Gr (nrXN)
H( )  a r ( ) W( )a Tt ( )
H( )  G r ( ) W ( )G Tt ( )
 
a r,i, j  exp(  jk j  rri )

G r,i, j  G r,i (k j )
Towards a stochastic MIMO channel model
 specification of Tx and Rx antennas, either through steering matrices
(uncoupled sensors) or through complex gain matrices
Towards a stochastic MIMO channel model
 specification of Tx and Rx antennas, either through steering matrices
(uncoupled sensors) or through complex gain matrices
 double directional model of emitted and received waves, specifying the
statistical laws of angular distributions on both sides of the radio link
Towards a stochastic MIMO channel model
 specification of Tx and Rx antennas, either through steering matrices
(uncoupled sensors) or through complex gain matrices
 double directional model of emitted and received waves, specifying the
statistical laws of angular distributions on both sides of the radio link
 statistical model for the wave connecting matrix , specifying the distribution
of complex entries of the matrix, especially the number of non zero entries for the
various columns or lines and their relative amplitudes.
Towards a stochastic MIMO channel model
 specification of Tx and Rx antennas, either through steering matrices
(uncoupled sensors) or through complex gain matrices
 double directional model of emitted and received waves, specifying the
statistical laws of angular distributions on both sides of the radio link
 statistical model for the wave connecting matrix , specifying the distribution
of complex entries of the matrix, especially the number of non zero entries for the
various columns or lines and their relative amplitudes.
 statistical model for the distribution of delays involved in the non zero entries
of W( )
MIMO channel model simplification
 Introduction of artificial junctions to reduce DOA/DOD number: depend on
maximum antenna size/angular resolution
Tx
Rx
MIMO channel model simplification
 Introduction of artificial junctions to reduce DOA/DOD number: depend on
maximum antenna size/angular resolution
Tx
Tx
Rx
Rx
MIMO channel model simplification
 Introduction of artificial junctions to reduce DOA/DOD number: depend on
maximum antenna size/angular resolution
Tx
Tx
Rx
Rx
 Limitation on the dynamic range of wave amplitudes: substitution of numerous
small amplitude waves by one or a few Rayleigh distributed waves of random
DOA/DOD.
Double directional channel measurements and junctions ?
• look for a differing number of DOAs and DODs
• look for several path delays for the same DOA (or DOD)
Y
X
Y
.
X
.
.
Conclusion
 Analysis of keyholes through Kirchhoff diffraction: continuous variation of
channel matrix effective rank with slit width
 Small antenna approximation yields a MIMO channel description entirely
based on DOA, DOD and antennas geometry
Junctions in multipath structure is responsible for the rectangular or non
diagonal character of the « wave connecting matrix »
 Coupling between sensors readily incorporated
 Stochastic channel model. Simplifications as a function of precision
requirements
 May feed simpler MIMO channel models with environment dependent
channel correlation matrices